Weak convergence

Weak convergence is a concept in probability theory describing how a sequence of random variables' distributions get closer to a specific distribution, in a way that focuses on their overall behavior rather than exact values. Imagine observing many outcomes that, over time, resemble a certain pattern or shape, even if individual results still vary. In essence, the distributions of these variables "converge weakly" to a target distribution, meaning their cumulative probabilities align more closely as the sequence progresses, providing a useful way to analyze approximate randomness or long-term behavior without requiring precise pointwise agreement.